Using subdivision surface cuts up my object?
Without seeing your blend file I can only make an assumption, but it appears as if you have a double-vertex edge on your cylinders. I can create a similar look by creating a circle, deleting half, duplicating it, rotating the duplicated half 180 degrees, joining them, extruding the shape, and then applying a subsurf.



Try going into edit mode, change to vertex mode (press 1), select all vertices by pressing A, right clicking and selecting Merge Vertices -> By Distance.



See if that resolves your cylinder issue.





Edit:

The issue is that you have an extra edge loop which has been folded over. You can either merge those vertices at both the top and bottom of each cylinder, or recreate the geometry with a better plan.